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46 Pages

### Sampling

Course: ECE 420, Fall 2009
School: Lake County
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Word Count: 1629

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of Sampling continuous-time signals Chap: 4.1-4.8; Pages: 140-201 Jont Allen ECE-310 Allen April 19, 2004 p.1/45 Periodic symmetry Every function may be made -periodic with an overlap and add OLA operation and integer, the period Functions periodic in one domain (e.g., time) are sampled in the other domain (e.g., frequency) Convergence of this expression is an issue Allen April 19, 2004...

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of Sampling continuous-time signals Chap: 4.1-4.8; Pages: 140-201 Jont Allen ECE-310 Allen April 19, 2004 p.1/45 Periodic symmetry Every function may be made -periodic with an overlap and add OLA operation and integer, the period Functions periodic in one domain (e.g., time) are sampled in the other domain (e.g., frequency) Convergence of this expression is an issue Allen April 19, 2004 p.2/45 Review of nomenclature FT type DTFT 48 time limits freq. limits in ROC z Transform 95 Fourier Transform 143 C/D Transform 143 Allen April 19, 2004 p.3/45 Periodic sampling 140 Starting from a continuous-time signal , a sampler determines a discrete-time signal C/D is sampled periodic in frequency Allen April 19, 2004 p.4/45 Periodic sampling Every periodic function may be expanded in harmonics, at frequencies From Fourier series formula: Periodic impulses: page 143, Eq. 4.5 Allen April 19, 2004 p.5/45 Basic symmetry Periodic impulses 143, Eq. 4.5 and Munson notes 25.2 This is the Poisson Summation Formula PSF PSF is a very important result Based on Fourier series expansion of impulse-train Allen April 19, 2004 p.6/45 Applications of PSF Let . Modulation formula: on right Multiply by on left, convolve Overlap-add formula: Convolve on left, multiply on right Allen April 19, 2004 p.7/45 Frequency domain nomenclature Details in working with and : A frequency of Hz has a radian frequency of , which corresponds to a normalized . frequency of Delta scaling for any an example. Example 4.1 148 . Try as Allen April 19, 2004 p.8/45 Pulse train modulation General case of time modulation Image Continuous time Fourier Transform pair Discrete time DTFT pair Allen April 19, 2004 p.9/45 Effect of increased sampling rate When is halved ( , doubled,): the images move out: Image Image Discrete time Discrete time 1X SAMPLING 2X SAMPLING Allen April 19, 2004 p.10/45 Effect of decreased sampling rate When is doubled ( , halved): the images move in. Overlap in the spectrum is called aliasing Image Image Discrete time Discrete time 1X SAMPLING 0.5X SAMPLING Allen April 19, 2004 p.11/45 Harry Nyquist Born in Sweden; Three famous theorems named after him: The Nyquist 1. Sampling Thm., (Nyquist 1928) 2. Thermal noise Thm., (Nyquist 1932) and 3. Feedback stability Thm. (Nyquist 1934) Allen April 19, 2004 p.12/45 Windowing the images What happens when images are removed by windowing in frequency? Image Continuous time Fourier Transform pair Discrete time DTFT pair Allen April 19, 2004 p.13/45 Nyquist sampling theorem 1928 Any signal may be uniquely represented by its samples if it is sampled at , dened as more than twice its highest frequency 146 Note the somewhat confusing denitions in the book 146 regarding the terms Nyquist frequency , versus the Nyquist rate . Allen April 19, 2004 p.14/45 Some issues to think about The proof of the Sampling Theorem is based on convolution with , namely the formula: 150 reconstructed This low-pass reconstruction lter is also called the interpolation lter, as it interpolates the signal between samples. Allen April 19, 2004 p.15/45 Some issues to think about In practice convolution by a perfect lter reconstructed is noncausal, and therefore cannot be implemented. A casual low-pass lter is used in practice. What are the practical implications of this? How will differ from the starting at the input to the ideal C/D followed by D/C conversion process? Namely what is the RMS error going to look like? In practice this works because the ear cannot hear the phase distortion Allen April 19, 2004 p.16/45 Poisson Summation Formula Case of impulse: General case of OLA comes from convolution on left by : is , sampled at is a continuous time function Allen April 19, 2004 p.17/45 Nyquists famous problem Find the Johnson thermal noise Transmission line terminated in resistors Length Stored energy Allen April 19, 2004 p.18/45 Nyquists 2 famous problem At , remove the resistors Length Stored energy Nyquists Johnson-noise formula follows: Allen April 19, 2004 p.19/45 DT processing of CT signals 4.4 Basic model of C/D D/C processing C/D/C 153-154 Discretetime system C/D D/C Ideal reconstruction (antialias) lter in D/C This basic structure describes almost every telephone, for more than 30 years Allen April 19, 2004 p.20/45 Two basic type of C/D/C systems There are two basic categories of C/D/C systems: Real-time processing: Any application where 1 sample in gives 1 sample out is a real-time method Non-real-time processing: non-real-time I applications are those where input time and output time are different, or non-real-time II where the computation time takes so much time that the processing cannot keep up Allen April 19, 2004 p.21/45 Real-time (RT) processing examples Today there are a great many examples of real-time systems: cell phones, CD players, hearing aids, video conferencing over a wide-band channel (i.e., the Internet) Sometimes we nd these systems to fall out of real time (i.e., cell phones and video) Allen April 19, 2004 p.22/45 Non-real-time (NRT) processing examples Cases of non-real-time applications where the input and output rates differ: Examples: pagers, fax machines, . . . These make you wait Allen April 19, 2004 p.23/45 Non-real-time processing examples II Cases of non-real-time applications where the computation time is greater than real-time video-conferencing over phone lines, cell-3G (3 generation cell), some classes of speech-De-noising and music encoding such as MPEG audio and video Allen April 19, 2004 p.24/45 OLA processing For many C/D/C processing schemes RT and NRT, OLA frequency domain processing is used This method is based on the OLA formula Let be a low-pass lter. small such that for . Under these conditions Allen April 19, 2004 p.25/45 OLA formula From the last slide: Typically , where is the length of window , Expand signal into OLA smooth blocks Dene the windowed signal as Allen April 19, 2004 p.26/45 From frequency to time by OLA Expand signal into smoothed OLA blocks As before: where Thus : Fourier Transform Allen April 19, 2004 p.27/45 Aliasing 4.1-4.3 147-149 Example of aliasing of a 110 Hz tone: Aliasing: every 11 sample of y[n] 1 y[n] th 0 1 0 2 1 0.2 F 100 50 0.4 0.6 Time [s] max 0 0.8 F max 1 110 [Hz] 10 0 150 50 100 150 Frequency [Hz] Sample period [s] [Hz]: [Hz] stems: Hz; Allen April 19, 2004 p.28/45 Aliasing 4.1-4.3 147-149 Example of decimation-aliasing of a tone: Aliasing: every 11 sample of y[n] 1 y[n] th 0 1 0 2 1 0.2 F 100 50 0.4 0.6 Time [s] max 0 0.8 F max 1 110 [Hz] 10 0 150 50 100 150 Frequency [Hz] Sample period [s] Hz; [Hz] For the blue curve . Thus , Allen April 19, 2004 p.29/45 Down-sampling 158 Suppose we cut the bandwidth by 2 in the frequency domain with an ideal low-pass lter We may then reduce at the output by 2x, otherwise without aliasing, thus Alternate samples are dropped in this processing, called down-sampling Allen April 19, 2004 p.30/45 Ideal differentiator 158 Suppose we wish to differentiate a continuous input signal This causal frequency response corresponds to Thus eff It may be shown that note the noncausal nature of Allen April 19, 2004 p.31/45 Impulse-invariance 160 Relate and to Discretetime system C/D D/C , eff Continuoustime LTI system With impulse invariance the mapping from discrete to the continuous domain is dene by Allen April 19, 2004 p.32/45 Example of impulse-invariance 162 A common continuous-time impulse response, and corresponding Laplace Transform: From impulse-invariance note error in book 4.8 162 This common example must alias since is not bandlimited Allen April 19, 2004 p.33/45 Upsampling by linear interpolation I When upsampling, we need to interpolate the new samples (Matlab help upsample, interp) 1 0 1 1 0 5 10 15 20 25 30 0 1 1 0 10 20 30 40 50 60 y[n+1] =(y[n]+y[n+2])/2 0 1 0.05 Error 0 10 20 30 40 50 60 0 0.05 0 10 20 30 40 50 60 This may be done by linear interpolation, but at a cost. Allen April 19, 2004 p.34/45 Upsampling by linear interpolation II Frequency response of a linear interpolator 1 0 1 1 0 5 10 15 20 25 30 0 1 1 0 10 20 30 40 50 60 y[n+1] =(y[n]+y[n+2])/2 0 1 0.05 0 10 20 30 40 50 60 Error 0 0.05 0 10 20 30 40 50 60 Even stems from linear interpolation Red curve is the exact; ): The error is of the form (note Allen April 19, 2004 p.35/45 Upsampling by linear interpolation III Distortion is audible F =1e4 sin(2 1000 n T) 10 10 10 10 10 10 20 s 0 f 10 3 0 Fs=10 4 20 20 10 4 lin interpolated 0 f 10 3 0 F =2 104 s F f s 0 10 4 20 error=|S(f)S (f)| i |error| = 5% 10 0 3 2 Fs=10 10 4 4 10 FREQUENCY (Hz) The error due to interpolation is 5% at There is an unwanted tone at Allen April 19, 2004 p.36/45 DT processing of analog signals 4.8 185 DT processing with A/D and D/A C/D Discretetime system D/C Sample and hold Anti aliasing filter A/D converter Discretetime system D/A converter Compensated reconstruction filter T T T Basic signal denitions Allen April 19, 2004 p.37/45 Traditional C/D conversion Traditional converter requires a high order lter [dB] Spectral noise floor Traditional filter [dB] Shaped noise floor Allen April 19, 2004 p.38/45 Some issues Analog prelter is very expensive, large, requires laser trimmed resistors Phase response distorts the waveform (not necessarily a problem) Needs to be a very large order (i.e., >100 dB/oct) The oversampled sigma-delta - converter solved all these proble...

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